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Group automorphisms with few and with many periodic points. (English) Zbl 1052.37017

Let \(T:X\rightarrow X\) be a compact group automorphism, \(F_n (T)=| \{ x\in X\mid T^n (x)=x\} | \) be the number of points of period \(n\). The main result of the paper is the following: for all \(C\in [0,\infty ]\) there is a compact group automorphism \(T\) with \(F_n (T)<\infty\) for all \(n\), and with \(\frac{1}{n}\log F_n (T)\rightarrow C\). This result is in contrast to the conjectured answer ‘yes’ in the open problem: \(\inf\{ h_{\text{top}} (T)\mid h_{\text{top}} (T)>0\} >0\)? (where \(h_{\text{top}} (T)\) is the topological entropy of \(T\)).

MSC:

37C35 Orbit growth in dynamical systems
22D40 Ergodic theory on groups
37B40 Topological entropy
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